regressiomitoitus

Regressiomitoitus is the process of estimating the parameters of a regression model from observed data. It is a core activity in regression analysis, aiming to determine numerical values for the model’s coefficients that best capture the relationship between a dependent variable and one or more independent variables.

In the common linear regression setting, Y = β0 + β1 X1 + ... + βk Xk + ε, regressiomitoitus seeks estimates of β0, β1, ..., βk and the error variance. The most widely used method is ordinary least squares (OLS), which finds coefficients that minimize the sum of squared residuals. Under standard Gauss–Markov assumptions (linearity, independence, homoscedasticity, correct specification), OLS estimators are unbiased and have the smallest variance among linear unbiased estimators.

Beyond OLS, estimation can be performed by maximum likelihood estimation (MLE), generalized least squares (GLS) when error terms are correlated or heteroscedastic, and regularized procedures such as ridge regression and lasso when predictors are numerous or highly collinear. Nonlinear regression uses nonlinear relationships and is estimated by nonlinear least squares or iterative algorithms.

Diagnostics and reporting include standard errors, t- or z-statistics, confidence intervals, p-values, and measures of fit such as R^2. Model assumptions are checked through residual analysis, plots, and formal tests; violations may require transformations, different estimators, or robust methods.

Regressiomitoitus also encompasses extensions such as generalized linear models (GLMs), mixed-effects models, and Bayesian estimation. The technique underpins both inference about relationships and making predictions on new data. Common challenges include multicollinearity, overfitting, heteroscedasticity, and autocorrelation, which can be mitigated with regularization, robust estimation, or cross-validation.